To understand how the brain functions, we must investigate the transient interactions that underpin communication between cortical regions. EEG possesses the optimal temporal resolution to capture functional connectivity, but it lacks the spatial resolution to identify the cortical locations responsible. To circumvent this problem electrophysiological connectivity should be investigated at the source level. There are many quantifiers of connectivity applied to EEG data, but some are not sensitive to the direct, or indirect, influence of one region over another, and others require the specification of a priori models so are unsuitable for exploratory analyses. Transfer Entropy (TE) can be used to infer the direction of linear and non-linear information exchange between signals over a range of time-delays within EEG data. This thesis explores the creation of a new method of mapping dynamic brain connectivity using a trial-based TE analysis of EEG source data, and the application of this technique to the investigation of semantic and number processing within the brain. The first paper (Chapter 2) documents the analyses of a semantic category and number magnitude judgement task using traditional ERP techniques. As predicted, the well-known semantic N400 component was found, and localised to left ATL and inferior frontal cortex. An N365 component related to number magnitude judgement was localised to right superior parietal regions including the IPS. These results offer support for the hub-and-spoke model of semantics, and the triple parietal model of number processing. The second paper (Chapter 3) documents an analysis of the same data with the new trial-based TE analysis. Word and number data were analysed at 0-200ms, 200-400ms, and 400-600ms following stimulus presentation. In the earliest window, information exchange was occurring predominately between occipital sources, but by the latest window it had become spread out across the brain. Task-dependent differences of regional information exchange revealed that temporal sources were sending more information to occipital sources following words at 0-200ms. Furthermore, the direction and timing of information movement within a front-temporal-parietal network was identified during 0-400ms of the number magnitude judgment. The final paper (Chapter 4), documents an attempt to track the influence of TMS through the brain using the TE analysis. TMS was applied to bilateral ATL and IPS because they are both important hubs in the brain networks that support semantic and number processing respectively. Left ATL TMS influenced sources located primarily in wide-spread left temporal lobe, and inferior frontal and inferior occipital cortices. The anatomical connectivity profile of the temporal lobe suggests that these are all plausible locations, and they exhibited excellent spatial similarities to the results of neuroimaging experiments that probed semantic knowledge. The analysis of right ATL TMS obtained a mirror image of the left. Left parietal stimulation resulted in a bilateral parietal, superior occipital, and superior prefrontal influence, which extended slightly further in the ipsilateral hemisphere to stimulation site. A result made possible by the short association and callosal fibres that connect these areas. Again, the results at the contralateral site were a virtual mirror image. The thesis concludes with a review of the experimental findings, and a discussion of methodological issues still to be resolved, ideas for extensions to the method, and the broader implications of the method on connectivity research.